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   pdfauthor={袁磊祺},
   pdftitle={波动变形壁面对湍流对流中热对流的增强}
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\title{波动变形壁面对湍流对流中传热的增强}
\author{袁磊祺，邹舒帆，杨延涛，陈十一} 
\institute{北京大学工学院力学与工程科学系}
\date{\today} 
\titlegraphic{\vspace{-0.3cm}\hfill\includegraphics[scale=0.18]{pkured.pdf}} 

\begin{document}

{
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\maketitle
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\metroset{titleformat frame=smallcaps}

\begin{frame}{目录}
	\setbeamertemplate{section in toc}[sections numbered]
	\tableofcontents
\end{frame}


\section{背景}

\begin{frame}[fragile]{湍流热对流}
	\vspace*{1em}
	湍流热对流在许多自然环境（例如地幔、太阳中的对流），和工程应用（工艺技术，金属生产）中起着至关重要的作用 (\cite{ahlers2009heat-transfer-a}).
	\begin{figure}
		\centering
		\begin{subfigure}[b]{0.24\textwidth}
			\centering
			\includegraphics[width=\textwidth]{Whole-mantle_convection}
			\caption{地幔对流。}
			\label{fig:mantle_convection}
		\end{subfigure}
		\hfill
		\begin{subfigure}[b]{0.24\textwidth}
			\centering
			\includegraphics[width=\textwidth]{global_ocean_conveyor}
			\caption{海洋环流。}
			\label{fig:global_ocean_conveyor}
		\end{subfigure}
		\hfill
		\begin{subfigure}[b]{0.24\textwidth}
			\centering
			\includegraphics[width=\textwidth]{cpu}
			\caption{CPU对流散热。}
			\label{fig:cpu}
		\end{subfigure}
		\hfill
		\begin{subfigure}[b]{0.24\textwidth}
			\centering
			\includegraphics[width=\textwidth]{home}
			\caption{室内对流.}
			\label{fig:home}
		\end{subfigure}
		\caption{湍流热对流.}
		\label{fig:Turbulence_convection}
	\end{figure}
\end{frame}


\begin{frame}[fragile]{Rayleigh-B\'{e}nard(RB)热对流系统}
	% \metroset{block=fill}

	\begin{block}{控制无量纲参数}
		\begin{itemize}
			\item $Ra= \frac{\alpha g \Delta H^3}{\nu\kappa}.$ 浮力比耗散力。
			\item $Pr=\nu / \kappa.$ 动量扩散比热扩散。
		\end{itemize}
	\end{block}
	\begin{block}{热传递效率$Nu$与控制参数间的标度率}
		\begin{itemize}
			\item $Nu=\frac{\left\langle w T\right\rangle_m - \kappa \left\langle \partial_z T \right\rangle_m}{\kappa \Delta H^{-1}} \sim Ra^{\gamma}Pr^{\beta}.$ 对流传热比热传导传热。
			\item $Nu \sim Ra^{1/3}$. (\cite{malkus1954the-heat-transp})
			\item $Nu \sim Ra^{1/2}Pr^{1/2}.$ 终极区间。(\cite{kraichnan1962turbulent-therm})
			\item GL 理论 (\cite{grossmann2001}).\vspace{1em}
			      $\tiny
				      \begin{cases}
					      Nu\, Ra\, Pr^{-2}=c_1 \frac{{Re}^2}{g\left(\sqrt{{Re}_c / {Re}}\right)}+c_2 {Re}^3 \\
					      {Nu}=c_3 {Re}^{1 / 2} {Pr}^{1 / 2}\left\{f\left[\frac{{Nu}}{2 \sqrt{{Re}_c}} g\left(\sqrt{\frac{{Re}_c}{{Re}}}\right)\right]\right\}^{1 / 2} +c_4 {Pr}\,  {Re}  \cdot f\left[\frac{{Nu}}{2 \sqrt{{Re}_c}} g\left(\sqrt{\frac{{Re}_c}{{Re}}}\right)\right] .
				      \end{cases}
			      $
		\end{itemize}
	\end{block}
	\begin{textblock*}{5cm}(10cm,2cm) % {block width} (coords)
		\begin{figure}[htp]
			\centering
			\includegraphics[width=6cm]{RBC.jpg}
			\caption{热对流系统。(\cite{xia2013current-trends-})}
			\label{fig:RBC}
		\end{figure}
	\end{textblock*}
\end{frame}

\begin{frame}[fragile]{静止粗糙壁面}
	\begin{columns}
		\begin{column}{0.6\linewidth}
			\begin{figure}[htp]
				\centering
				\includegraphics[width=0.7\textwidth]{fig/xie_xia.png}
				\caption{金字塔。(\cite{xieTurbulentThermalConvection2017})}
				\label{fig:xie_xia}
			\end{figure}
			\vspace{-2em}
			\begin{figure}[htp]
				\centering
				\includegraphics[width=0.4\textwidth]{fig/emran.png}
				\caption{环形. (\cite{emran_shishkina_2020})}
				\label{fig:emran}
			\end{figure}
		\end{column}
		\begin{column}{0.4\linewidth}
			\begin{figure}[htp]
				\centering
				\includegraphics[width=0.65\textwidth]{zhuRoughnessFacilitatedLocalScaling2017.png}
				\caption{正弦函数。(\cite{zhuRoughnessFacilitatedLocalScaling2017})}
				\label{fig:zhuRoughnessFacilitatedLocalScaling2017}
			\end{figure}
		\end{column}
	\end{columns}
\end{frame}

\begin{frame}[fragile]{调制方法}
	\begin{columns}
		% \hspace{-2em}
		\begin{column}{0.6\linewidth}
			\begin{figure}[htp]
				\centering
				\includegraphics[width=0.7\textwidth]{fig/wang2020vibration-induc.jpg}
				\caption{水平体震动(\cite{wang2020vibration-induc})。 $\partial_t \bm{u}+(\bm{u} \cdot \nabla) \bm{u}=-\nabla p+\nu \nabla^2 \bm{u}+\alpha T\left[g \vec{z}+{\color{PKUred}A \Omega^2 \cos (\Omega t) \vec{x}}\right]$.}
				\label{fig:wang2020vibration}
			\end{figure}
		\end{column}
		% \hspace{1em}
		\begin{column}{0.4\linewidth}
			\begin{figure}[htp]
				\centering
				\includegraphics[width=0.65\textwidth]{yang2020periodically-mo}
				\caption{边界温度周期变化(\cite{yang2020periodically-mo}). $T_{\mathrm{bot}}=1+A \cos (2 \pi f t)$.}
				\label{fig:yang2020periodically-mo}
			\end{figure}
		\end{column}
	\end{columns}
\end{frame}

\section{控制方程和数值方法}

\begin{frame}[fragile]{控制方程，几何外形和参数空间}
	\begin{columns}
		% \hspace{-2em}
		\begin{column}{0.5\linewidth}
			\begin{equation*}
				\left\{
				\begin{aligned}
					\frac{\partial \bm{u}}{\partial t}+\bm{u} \cdot \nabla \bm{u} & =-\frac{1}{\rho} \nabla p+ \nu \nabla^{2} \bm{u}+\alpha T g \bm{e}_{z}, \\
					\frac{\partial T}{\partial t}+\bm{u} \cdot \nabla T           & =\kappa \nabla^{2} T,                                                   \\
					\nabla \cdot \bm{u}                                           & =0.
				\end{aligned}
				\right.
			\end{equation*}
			% \vspace{2em}
			\begin{itemize} %The symbol of the items can be changed by which ever you want, this is just an example.
				\item 瑞利数$Ra:  10^{6} \sim 10^{8}$.
				\item 普朗特数$Pr=1$.
				\item 振幅$A / H: 0.02 \sim 0.08$.
				\item 波数 $kH: 0.5 \sim 8$.
				\item 频率$f\tau: 0.2 \sim 8.$
			\end{itemize}
		\end{column}
		\hspace{1em}
		\begin{column}{0.48\linewidth}
			\begin{figure}
				\centering
				\includegraphics[height=0.35\linewidth]{fig-geometry.pdf}\\
				\caption{$z(x,y,t)=z_0 - A \cos( 2\pi k x) \cos( 2\pi k y) \cos( 2\pi f t)$.}
			\end{figure}
			\vspace{-1em}
			\begin{figure}
				\centering
				\includegraphics[width=0.7\linewidth]{temperature3D.png}\\
				\caption{温度. $Ra = 10^7,\, kH = 4,\, f\tau = 4,\, A/H = 0.08$.}
			\end{figure}
		\end{column}
	\end{columns}
\end{frame}

\begin{frame}[fragile]{浸入边界法(IBM)}
	\begin{columns}
		\begin{column}{0.4\linewidth}
			\begin{block}{优点}
				\vspace*{-2pt}
				网格生成简单，为笛卡尔网格,处理运动边界很方便。
			\end{block}
			\begin{block}{缺点}
				\vspace*{-2pt}
				不方便局部加密，很难处理复杂外形高雷诺数的情况。
			\end{block}
		\end{column}
		\hspace{1em}
		\begin{column}{0.5\linewidth}
			\begin{figure}[ht]
				\includegraphics[width=\linewidth]{ibm1}\\
				\caption{网格划分。(\cite{mittal2005immersed-bounda})}
			\end{figure}
		\end{column}
	\end{columns}
\end{frame}


\section{热对流增强和流动调制}

\begin{frame}[fragile]{动画}
\end{frame}

\begin{frame}[fragile]{$Nu,\,Re$随$k$的变化}
	\begin{columns}
		\begin{column}{0.6\linewidth}
			\begin{itemize}
				\item $Nu$最大增长超过 75\%. $Re = u_{\rm{rms}}H / \nu$ 变化不明显。
				\item $Re$ 变化不明显。
				\item 振幅和边界层厚度关系可能是决定波动壁面能否增强热对流的重要因素.
			\end{itemize}
		\end{column}
		\begin{column}{0.4\linewidth}
			\begin{figure}
				\centering
				\includegraphics[width=0.85\linewidth]{figure1.eps}\\
				\caption{(a-b) $f\tau=1$, 变化 $(Ra, A/H)$。(c)速度温度边界层厚度。}
			\end{figure}
		\end{column}
	\end{columns}
\end{frame}

\begin{frame}[fragile]{波数 $k$ 对流场的影响}
	\begin{columns}
		\begin{column}{0.5\linewidth}
			\begin{itemize}
				\item $kH=0.5$时流体贴合壁面运动，影响对流区的运动。
				\item $kH=8$ 时，震动主要影响壁面附近，对流区域还是由热羽流驱动。
				\item 壁面热流主要来自波峰处。
			\end{itemize}
		\end{column}
		\begin{column}{0.5\linewidth}
			\begin{figure}
				\centering
				\includegraphics[width=0.7\linewidth]{figure2.pdf}
				\caption{$Ra=10^7,\,A/H=0.08,\,f\tau=1$. (a)(c)温度， (b)(d)法向速度。(e)半个波长壁面上的温度梯度。}
			\end{figure}
		\end{column}
	\end{columns}
\end{frame}


\begin{frame}[fragile]{$Nu,\,Re$随 $f$ 的影响}
	\begin{columns}
		\begin{column}{0.58\linewidth}
			\begin{itemize}
				\item 固定 $kH=4$. $Nu$随$f$先减小，然后再迅速增长，最高增长 100\%.
				\item $Nu$的增长伴随着$Re$的增长。
				\item 用 $Nu_{\mathrm{min}}$ 和对应的 $f_{\mathrm{min}}$ 归一化后，曲线大致重合。
			\end{itemize}
		\end{column}
		\begin{column}{0.4\linewidth}
			\begin{figure}
				\centering
				\includegraphics[width=0.75\linewidth]{figure3.eps}
				\caption{(a) $Nu$. (b) $Re$. (c)归一化后的 $Nu$.}
			\end{figure}
		\end{column}
	\end{columns}
\end{frame}

\begin{frame}[fragile]{壁面流场随 $f$ 的变化}
	\begin{itemize}
		\item 随着$f$的增长，两个大涡先往下移动到靠近波谷的位置，然后分成多个涡。
		\item 壁面温度梯度非线性变化。
	\end{itemize}
	\begin{figure}
		\centering
		\includegraphics[width=0.99\linewidth]{figure4.pdf}\\
		\caption{(a-c) 温度场和平均流线。$f\tau=0.2,1,5$. (d) 壁面温度梯度。}
	\end{figure}
\end{frame}


\begin{frame}[fragile]{增强传热效果}
	\begin{figure}
		\centering
		\includegraphics[width=0.65\linewidth]{figure5.eps}
		\caption{红色点为$A/H=0.04$, $kH=4$,  $f\tau=8$. 蓝色点为标准RB算例，灰色虚线表示 $Nu \sim Ra^{0.3 \pm 0.03}$ 标度率。}
	\end{figure}
\end{frame}


\begin{frame}[fragile]{壁面面积和热流增强}
	\begin{figure}
		\centering
		\includegraphics[width=0.9\linewidth]{fig-AreaRatio.eps}
		\caption{(a) $S$: 归一化后的面积，虚线为静止壁面，实线为时间平均的动边界. (b-c) $S_{\mathrm{avg}}$: 归一化后的时间平均动边界面积.}
	\end{figure}
\end{frame}

\begin{frame}[fragile]{温度耗散}
	\begin{equation}
		\epsilon_T=\langle \kappa (\nabla T)^2 \rangle_{V,t} = \kappa \Delta^2 H^{-2} Nu.
	\end{equation}
	\begin{figure}
		\centering
		\includegraphics[width=0.7\linewidth]{fig-Nuhist.eps}\\
		\caption{温度耗散的时间序列。不同的曲线表示不同的$(kH,f\tau\,)$组合。$Ra=10^7,A/H=0.08$.}
	\end{figure}
\end{frame}

\begin{frame}[fragile]{动能耗散和壁面变形}
	\begin{equation}
		\left<\int_V \Phi  \dif V \right>_t= \left<-\oint_{\partial V} (\bm{u}p) \cdot \dif \bm{s}  \right>_t +  \left< \mu \oint_{\partial V} (\bm{u} \cdot 2 \bm{S} ) \cdot \dif \bm{s} \right>_t + (Nu -1) \frac{\kappa \Delta \alpha g \rho}{H}
		\label{eq:wVeloSV}
	\end{equation}
	\begin{itemize}
		\item 定义 $E_k =  \frac{H}{\kappa \Delta \alpha g \rho} \left[ \left<-\oint_{\partial V} (\bm{u}p) \cdot \mathrm{d}  \bm{s}  \right>_t +  \left< \mu \oint_{\partial V} (\bm{u} \cdot 2 \bm{S} ) \cdot \mathrm{d} \bm{s} \right>_t \right].$
	\end{itemize}
	\begin{figure}
		\centering
		\includegraphics[width=0.8\linewidth]{fig-veloDisspVsNuF1.eps}
		\caption{不同的符号表示不同的$(Ra,A/H)$组合.}
	\end{figure}
\end{frame}



\section{结论}

\begin{frame}[fragile]{结论}
	\begin{itemize}
		\setlength{\itemsep}{10pt}
		\item 当振幅超过边界层厚度时，增加波数可以获得较为明显的热对流增强.
		\item 频率增长到某一值后， $Nu$能快速增长，并能改变波谷内的流动结构和增强整体的流动强度。
		\item 当震动壁面参数固定时，由于震动壁面可以有效地打破限制热对流的边界层， $Nu$ 比 $Ra$ 的标度率可以趋于终极态的 0.5.
	\end{itemize}
\end{frame}

\appendix

\begin{frame}
	\Huge
	\center
	谢谢！欢迎批评指正!
\end{frame}



\begin{frame}[allowframebreaks]{参考文献}
	\printbibliography[heading=none]
\end{frame}


\end{document}
